Cubic Forms - Algebra, Geometry, Arithmetic

It is proved that the commutator ideal of the multiplication algebra of a free commutative alternative algebra of rank n is nilpotent of index n − 1. As a corollary to this fact, the Bruck theorem for special commutative Moufang loops is derived.

CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.

Let [Formula: see text] be a smooth projective rational variety carrying a regular action of a finite abelian group [Formula: see text]. We give examples of effective computation of the Brauer group of the quotient stack [Formula: see text] in dimensions [Formula: see text] and [Formula: see text] using residues in Galois cohomology and the geometry of fixed loci. In particular, we compute [Formula: see text] for all [Formula: see text]-minimal del Pezzo surfaces.

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

Topological interface modes in 3D-printed triply periodic minimal surface phononic crystals

Let k be a perfect field with cohomological dimension [les ] 2. Serre's conjecture II claims that the Galois cohomology set H 1 ( k , G ) is trivial for any simply connected semi-simple algebraic G / k and this conjecture is known for groups of type 1 A n after Merkurjev–Suslin and for classical groups and groups of type F 4 and G 2 after Bayer–Parimala. For any maximal torus T of G / k , we study the map H 1 ( k , T ) → H 1 ( k , G ) using an induction process on the type of the groups, and it yields conjecture II for all quasi-split simply connected absolutely almost k -simple groups with type distinct from E 8 . We also have partial results for E 8 and for some twisted forms of simply connected quasi-split groups. In particular, this method gives a new proof of Hasse principle for quasi-split groups over number fields including the E 8 -case, which is based on the Galois cohomology of maximal tori of such groups.

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

A commutative Moufang loop which is not associative has the property that all proper subloops are associative if and only if the loop can be generated by three elements.

Algebraic commutative Moufang loops

We present an asymptotic analysis of shell lattice metamaterials based on Ciarlet's shell theory, introducing a new metric— asymptotic directional stiffness (ADS)—to quantify how the geometry of the middle surface governs the effective stiffness. We prove a convergence theorem that rigorously characterizes ADS and establishes its upper bound, along with necessary and sufficient condition for achieving it. As a key result, our theory provides the first rigorous explanation for the high bulk modulus observed in Triply Periodic Minimal Surfaces (TPMS)-based shell lattices. To optimize ADS on general periodic surfaces, we propose a triangular-mesh-based discretization and shape optimization framework. Numerical experiments validate the theoretical findings and demonstrate the effectiveness of the optimization under various design objectives. Our implementation is available at https://github.com/lavenklau/minisurf.

Hydroxyapatite-Wollastonite/Magnesium alloy (HAp-Wol/Mg3Zn) Interpenetrating Phase Composites (IPCs) were developed by combining two different fabrication technologies, giving rise to a hybrid route for manufacturing innovative biomedical composites. A triply periodic minimal surface (TPMS) geometry was modelled, exploring two different volume fractions to control ceramic/metal ratio and interconnectivity. HAp-Wol structures were manufactured using additive manufacturing vat photopolymerization, specifically Digital Light Processing technology. The thermal debinding and sintering process of the printed HAp-Wol ceramic were optimized to allow a subsequent infiltration of the TPMS structures with a Mg3Zn alloy via investment casting, materialising the designed interpenetrating phase composites (IPCs). This approach enabled an effective infiltration of the alloy without the presence of major defects. The compressive strength of the HAp-Wol/Mg3Zn IPCs was significantly higher than that of the ceramic counterparts, increasing from 6.5 MPa to 162.5 MPa for the IPCs having the lowest ceramic ratio. The ceramic phase evidenced the formation of an apatite-like layer at their surface upon in vitro dissolution tests, evidencing bioactivity, being the calcium-rich silicate phase strongly responsible for this behaviour. IPCs revealed a higher dissolution rate than the ceramic counterpart. This study demonstrates the feasibility of this hybrid manufacturing route to fabricate HAp-Wol/Mg3Zn IPCs that can be tailored by design to meet the requirements of bone substitutes, namely the mechanical performance and absorption rate.

We initiate a systematic scan of the landscape of black holes in any spacetime dimension using the recently proposed blackfold effective worldvolume theory. We focus primarily on asymptotically flat stationary vacuum solutions, where we uncover large classes of new black holes. These include helical black strings and black rings, black odd-spheres, for which the horizon is a product of a large and a small sphere, and non-uniform black cylinders. More exotic possibilities are also outlined. The blackfold description recovers correctly the ultraspinning Myers-Perry black holes as ellipsoidal even-ball configurations where the velocity field approaches the speed of light at the boundary of the ball. Helical black ring solutions provide the first instance of asymptotically flat black holes in more than four dimensions with a single spatial U(1) isometry. They also imply infinite rational non-uniqueness in ultraspinning regimes, where they maximize the entropy among all stationary single-horizon solutions. Moreover, static blackfolds are possible with the geometry of minimal surfaces. The absence of compact embedded minimal surfaces in Euclidean space is consistent with the uniqueness theorem of static black holes.

We study complete minimal surfaces in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy W:=14∫|H→|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}: =\\frac{1}{4} \\int |\\vec H|^2$$\\end{document}. In codimension one, we prove that the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly m-3=W4π-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-3=\\frac{\\mathcal {W}}{4\\pi }-3$$\\end{document}. We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index of their inverted surfaces.

Minimal surfaces have been gradually translated from mathematics to architectural design research due to their fascinating geometric and spatial properties.Tensile structures are just an example of their application in architecture known since the early 1960s.The present research relates to the problem of generating minimal surface geometries computationally using self-organizing particle spring systems and optimizing them for digital fabrication.The algorithm is iterative and it has a different approach than a standard computational method, such as dynamic relaxation, because it does not start with a pre-defined topology and it consists of simultaneous processes that control the geometry' s tessellation.The method is tested on triply periodic minimal surfaces and focused on several fabrication techniques such as a tensegrity modular system composed of interlocked rings (Figure 1).paper session | Minimal Surfaces as Self-organizing Systems